A325011 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

1, 0, 1, 4, 3, 0, 1, 8, 28, 36, 15, 0, 1, 13, 84, 282, 465, 360, 105, 0, 1, 19, 192, 1110, 3711, 7080, 7560, 4200, 945, 0, 1, 26, 381, 3320, 17875, 60159, 126728, 165900, 130725, 56700, 10395, 0, 1, 34, 687, 8484, 66525, 340929, 1158102, 2624748, 3964905, 3931200, 2453220, 873180, 135135, 0

Offset: 1

Author

Robert A. Russell, May 27 2019

Keywords

• nonn
• tabf
• easy

Comments

Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. An achiral coloring is identical to its reflection.

Also the number of achiral colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors.

Examples

			Table begins with T(1,1):
 1  0
 1  4   3    0
 1  8  28   36    15     0
 1 13  84  282   465   360    105      0
 1 19 192 1110  3711  7080   7560   4200    945     0
 1 26 381 3320 17875 60159 126728 165900 130725 56700 10395 0
For T(2,3)=3, each of the three chiral pairs has two opposite edges with the same color.
		

Links

• Robert A. Russell, Table of n, a(n) for n = 1..132

Crossrefs

Cf. A325008 (oriented), A325009 (unoriented), A325010 (chiral), A325007 (up to k colors).

Other n-dimensional polytopes: A325003 (simplex), A325019 (orthoplex).

Programs

Mathematica

Table[Sum[Binomial[-j-2,k-j-1] Binomial[n + Binomial[j+2,2]-1, n], {j,0,k-1}] - Sum[Binomial[j-k-1,j] Binomial[Binomial[k-j,2],n],{j,0,k-2}], {n,1,10},{k,1,2n}] // Flatten

Formula

T(n,k) = Sum_{j=0..k-1} binomial(-j-2,k-j-1) * binomial(n + binomial(j+2,2)-1, n) - Sum_{j=0..k-2} binomial(j-k-1,j) * binomial(binomial(k-j,2),n).

T(n,k) = 2*A325009(n,k) - A325008(n,k) = A325008(n,k) - 2*A325010(n,k) = A325009(n,k) - A325010(n,k).